Within-class scatter matrixIt is used to represent the distribution of sample points around the mean, and its definition is as follows:

Suppose there are $latex {M}$ categories, $latex {Ω\mathop{{}}\nolimits_{{i}},…,Ω\mathop{{}}\nolimits_{{M}}}$ , $latex {Ω\mathop{{}}\nolimits_{{i}}}$ class sample set$latex { \left\{ {X\mathop{{}}\nolimits_{{1}}^{{{ \left( {i} \right) }}},X\mathop{{}}\nolimits_{{2}}^{{{ \left( {i} \right) }}},…,X\mathop{{}}\nolimits_{{N\mathop{{}}\nolimits_{{i}}}}^{{{ \left( {i} \right) }}}} \right\} }$ , $latex {Ω\mathop{{}}\nolimits_{{i}}}$ The divergence matrix of the class is defined as:

$latex {S\mathop{{}}\nolimits_{{w}}^{{{ \left( {i} \right) }}}\text{ }=\text{ }\frac{{1}}{{N\mathop{{}}\nolimits_{{i}}}}{\mathop{ \sum }\limits_{{k=1}}^{{N\mathop{{}}\nolimits_{{i}}}}{{ \left( { {X\mathop{{}}\nolimits_{{k}}^{{{ \left( {i} \right) }}}-m\mathop{{}}\nolimits^{{{ \left( {i} \right) }}}} \right) }\mathop{{}}\nolimits^{{T}}}}}$

Among them, $latex {S\mathop{{}}\nolimits_{{{w}}}^{{ \left( {i} \right) }}}$ is the covariance matrix of the class $latex {Ω\mathop{{}}\nolimits_{{i}}}$.

The total intra-class scatter matrix is:

$latex {S\mathop{{}}\nolimits_{{w}}\text{ }=\text{ }{\mathop{ \sum }\limits_{{i=1}}^{{M}}{P{ \left( {Ω\mathop{{}}\nolimits_{{i}}} \right) }S\mathop{{}}\nolimits_{{w}}^{{{ \left( {i} \right) }}}}}\text{ }=\text{ }{\mathop{ \sum }\limits_{{i=1}}^{{M}}{P{ \left( {Ω\mathop{{}}\nolimits_{{i}}} \right) }\frac{{1}}{{N\mathop{{}}\nolimits_{{i}}}}{\mathop{ \sum }\limits_{{k=1}}^{{N\mathop{{}}\nolimits_{{i}}}}{{ \left( { {X\mathop{{}}\nolimits_{{k}}^{{{ \left( {i} \right) }}}-m\mathop{{}}\nolimits^{{{ \left( {i} \right) }}}} \right) }\mathop{{}}\nolimits^{{T}}}}}}}$

Then: trace$latex { \left\{ {S\mathop{{}}\nolimits_{{w}}} \right\} }$ is the average measure of feature variance of all classes.

Regarding the results of feature selection and extraction, the smaller the product of the within-class scatter matrix, the better.

Related words: between-class scatter matrix